November 2000 Volume 3 Number Supp pp 1192 - 1198
Computational approaches to
sensorimotor transformations
Alexandre Pouget1 & Lawrence H. Snyder2
1. Department of Brain and Cognitive Sciences, University of
Rochester, Rochester, New York 14627, USA
2. Department of Anatomy and Neurobiology, Washington University,
660 S. Euclid Avenue, St. Louis, Missouri 63110, USA
Correspondence should be addressed to A Pouget.
alex@bcs.rochester.edu
Behaviors such as sensing an object and then moving your
eyes or your hand toward it require that sensory information
be used to help generate a motor command, a process known
as a sensorimotor transformation. Here we review models of
sensorimotor transformations that use a flexible intermediate
representation that relies on basis functions. The use of basis
functions as an intermediate is borrowed from the theory of
nonlinear function approximation. We show that this approach
provides a unifying insight into the neural basis of three
crucial aspects of sensorimotor transformations, namely,
computation, learning and short-term memory. This
mathematical formalism is consistent with the responses of
cortical neurons and provides a fresh perspective on the issue
of frames of reference in spatial representations.
The term 'sensorimotor transformation' refers to the process by which
sensory stimuli are converted into motor commands. This process is
crucial to any biological organism or artificial system that possesses
the ability to react to the environment. Accordingly, this topic has
attracted considerable attention in neuroscience as well as
engineering over the last 30 years.
A typical example of such a transformation is reaching with the hand
toward a visual stimulus. In this case, as for most sensorimotor
transformations, two issues must be resolved. First, one must
determine the configuration of the arm that will bring the hand to the
spatial location of the visual stimulus (kinematics). The second
problem is specifying and controlling the application of force to
determine the movement trajectory (dynamics)1, 2. This review
focuses almost exclusively on kinematics (see Wolpert and
Ghahramani, this issue, for models of movement dynamics).
Our goal is to provide an overview of the basis function approach to
sensorimotor transformations. In this approach, sensory information
is recoded into a flexible intermediate representation to facilitate the
transformation into a motor command. This has the advantage of
explaining how the same neurons can be engaged in three seemingly
distinct aspects of sensorimotor transformations, namely,
computation, learning and short-term memory. We first review the
theory behind representing and transforming spatial information
using basis functions. Next we describe how these transformations
can be learned using biologically plausible algorithms. Finally, we
explain how to implement short-term spatial memory and updating of
motor plans using these representations. In each case, we examine
the extent to which models relying on basis functions are consistent
with known neurobiology. Remarkably, all three tasks—computation,
learning and short-term memory of spatial representations—can be
efficiently handled using a neural architecture derived from the basis
function approach. As we will see, a basis function representation is a
form of population code, and we argue that the exceptional
computational versatility of basis functions may explain why
population codes are so ubiquitous in the brain.
Basis functions for sensorimotor transformations
Sensorimotor transformations are often formalized in terms of
coordinate transformations. For instance, to reach for an object
currently in view, the brain must compute the changes in joint angles
of the arm that will bring the hand to the desired spatial location. This
computation requires combining visual information—the retinal or
eye-centered coordinates of the object—with signals related to the
posture of body parts, such as the position of the eyes in the head
(eye position), the position of the head with respect to the trunk
(head position) and the starting position of the arm. We refer to such
positional signals as 'posture signals'. In this way, we can recast a
coordinate transformation as the computation of the value of a
particular function. This function takes visual and postural signals as
input and produces as output the set of changes in joint angles
required to solve the task (for example, bring the hand to the target).
Recasting the coordinate transformation as computing the value of a
function makes it easier to generate and test biologically inspired
models of this aspect of brain function.
We will adopt a vectorial notation for the signals being transformed.
V is a vector that encodes an object's location in eye-centered space.
It has three components, which correspond, respectively, to the
image's azimuth and elevation on the retina and the object's distance
from the retina. This is clearly not the form of the representation
used by the brain, but the vector format is not important at this stage
(see below). We use similar notations, P and J, for the posture
signals and the change in joint coordinates. A coordinate transform
can then be written as a function, f(), mapping V and P onto J: J =
f(V, P).
It is useful to divide all functions into two classes, linear and
nonlinear (Fig. 1). Sensorimotor transformations almost exclusively
belong to the second class. The nonlinearity arises from the geometry
of our joints. The change in spatial location of the hand that results
from bending the elbow depends not only on the amplitude of the
elbow movement, but also on the state of the shoulder joint. As a
result, a neural network implementation of a sensorimotor
transformation requires at least three layers. There must be at least
one intermediate layer (the so-called 'hidden layer') to recode the
sensory inputs before they can be transformed into motor commands
(Box 1; Fig. 1d). One of the challenges in computational neuroscience
has been to identify intermediate representations that are both
biologically plausible and computationally efficient for these nonlinear
mappings.
One solution involves using intermediate units that compute basis
functions3-5, because most functions of interest can be approximated
using a linear combination of basis functions. The best known basis
set is that used by the Fourier transform: any function can be
expressed as the linear sum of a series of cosine and sine functions of
arbitrary amplitudes and frequencies. Many other functions can be
used to form basis sets (Box 1 ; Fig. 1d).
When applied to sensorimotor transformations, and in particular to
the example of reaching toward a visual target, the idea is that the
reaching motor command J can be obtained by taking a weighted
sum of N basis functions {Bi(V,P)}N i=1 of the visual and posture
signals, V and P:
The set of
weights, {wi} Ni=1, is specific to the reaching motor command being
computed and, as we will see later, can be determined using simple
learning rules.
Many choices are available for the basis functions. For instance, one
can use the set of all Gaussian functions of V and P, which are a
subset of a larger family known as radial basis functions (RBF) 3. The
network in Fig. 1d is an example of a radial basis function network in
which the variables considered are x and y instead of V and P. This
type of representation is also sometime called a population code, that
is, a code in which the variables are encoded through the activity of a
large population of neurons with overlapping bell-shape tuning
curves. Population codes may be ubiquitous in the nervous system
because they provide basis sets.
An alternative choice for a basis set is the product of a Gaussian
function of the eye-centered position of an object (V) and a sigmoid
function of eye position (P). In an idealized neuron that performs this
calculation (Fig. 2a), the receptive field is eye-centered, that is, it
remains at the same location relative to the fovea regardless of eye
position (Fig. 2b). However, response gain (that is, its amplitude)
changes with eye position.
From a biological point of view, one problem with Eq. 1 is the format
of the input and output vectors. For instance, we used polar
coordinates for vector V, yet no such vector has been explicitly
identified in the cortex. Instead, the visual position of objects is
encoded by the activity of a large number of binocular neurons
forming the retinotopic maps in the early visual areas. This does not
mean that we cannot use the basis function framework. We simply
replace the vector V with a new vector, VA, which has as many
components as there are neurons in the retinotopic map; each
component corresponds to the activity (for example, firing rate) of
one neuron. Likewise, the vectors P and J can be replaced by the
corresponding neuronal patterns of activities PA and JA (Fig. 2c ).
Many network models of sensorimotor transformations rely on such
basis function representations in their intermediate layer6-9.
Biological plausibility of the basis function approach
The basis function approach requires that the tuning curves of
neurons in intermediate stages of computation provide a basis
function set. A set of functions constitutes a basis set if certain
requirements are met. First, the functions must combine their inputs,
for example, the visual input, V, and the posture signals, P,
nonlinearly and so that they cannot be decomposed into separate
functions of V and functions of P. This rules out linear functions and
functions of the type Bi( V,P) = Ci(V) + Di(P). Furthermore, the
functions must be able to fully cover the range of possible input
values5. In other words, there must be units with all possible
combinations of selectivity for visual and posture signals.
Neurons whose response can be described by a Gaussian function of
retinal location multiplied by a sigmoidal function of eye position
would qualify (Fig. 2a and b)5. Many such gain-modulated neurons
are found in the parietal lobe, where there are neurons with all
possible combinations of visual and eye position selectivities10. Gain
modulations between sensory and posture signals are also observed
in occipital11-15 and premotor cortices16, suggesting that basis
function representations may be widely used.
In many early papers, gain modulation by posture signals was
reported to be linear, not sigmoidal. This is clearly incompatible with
the basis function hypothesis, as basis functions require nonlinear
tuning curves. These experiments, however, were designed to detect
an effect, not to distinguish the precise form of the gain field. A linear
model of gain fields was simple and lent itself most easily to
statistical testing. However, recent experiments17 and new analyses5
reveal significant nonlinearities consistent with sigmoidal modulation.
This conclusion is based on data from the parietal cortex, but given
the similarities among gain fields throughout the cortex, it is
reasonable to think that it applies to most gain fields.
Another line of evidence in support of the basis function approach
comes from the study of hemineglect patients with right parietal
cortex lesions. These patients tend to ignore sensory stimuli located
on their left18. 'Left', however, can be defined with respect to multiple
frames of reference; it could be the left side with respect to the eyes
(that is, the left visual field), head or body. For example, consider a
subject who turns his head to the left of a stimulus that lies directly in
front of him, but then moves his eyes far to the right. The stimulus
will now lie on the midline with respect to the body, to the right with
respect to the head, and to the left with respect to the eyes. By
assessing neglect using a variety of body postures and stimulus
locations, one can attempt to determine the relevant frame of
reference for a given patient's neglect. Interestingly, such
experiments show that neglect often affects multiple frames of
reference (for review, see ref. 19).
This observation fits well with one property of basis function
representations, namely, that they encode location in multiple frames
of reference simultaneously. For instance, a basis function
representation integrating a visual input with eye position signals
(Fig. 2c) represents the location of objects in eye- and head-centered
frames of reference simultaneously5. Indeed, to recover the position
of an object in, say, head-centered coordinates, one must compute a
function of the eye-centered position of the object as well as the
current eye and head positions. As for any other function, this can be
done with a simple linear transformation of the activity of the basis
function units. As a result, a simulated lesion of a basis function
representation can explain why hemineglect affects several frames of
reference across a variety of tasks19.
The multiplicity of frames of reference is one of the most
distinguishing properties of basis function representations. In more
traditional approaches to spatial representations, object position is
represented in maps using one particular frame of reference. Multiple
frames of reference require multiple maps, and a neuron can only
contribute to one frame of reference, specific to its map. By contrast,
in a basis function map, each neuron contributes to multiple frames
of reference. Thus, basis function neurons are ideally placed to
coordinate different behaviors, such as moving the eyes and hand to
the same object, even though these movements must be
programmed in distinct coordinates.
Learning sensorimotor transformations
A few sensorimotor transformations (such as the eyeblink reflex) may
already be wired at birth and require little training. In most cases,
however, the mapping from sensory to motor coordinates must be
learned and updated through life, as eyes, arms and other body parts
change in size and weight. As before, we focus exclusively on the
issue of coordinate transformations. How do we learn and maintain a
mapping of sensory coordinates of objects into motor coordinates?
Piaget20 proposed that babies learn by associating spontaneous motor
commands with the sensory consequences of those spontaneous
actions. Consider how this would apply to a two-layer network used
to control arm movements. We assume that the input layer encodes
the visual location of the hand, whereas the output layer represents
reaching motor commands in joint-centered coordinates. (Motor
commands actually require changes in joint angles, but for simplicity
we will consider absolute angles.) On each trial, the network
generates a spontaneous pattern of activity in the motor layer. This
pattern is fed to the arm, which moves accordingly, and the network
receives visual feedback of the resulting hand position. At this point,
the system can learn to associate the patterns in the sensory and
output layers. In particular, the Hebb rule can be used to increase the
weights between co-active sensory and motor units21.
A further refinement is to treat the position of the hand after the
spontaneous movement as a target to be reached. The idea is to first
compute the motor command that the network would have generated
if it had aimed for that location from the start. We call this the
'predicted' motor command. (Note that this movement is not actually
executed.) We can then compare this predicted command to the
original spontaneous one. Because the spontaneous command is
precisely the command that brings the hand to the current location,
we should adjust the network weights to make the predicted motor
command closer to the spontaneous one. To compute the predicted
motor command, we use the visually determined hand location after
the spontaneous movement as a network input, and use the current
weights to compute the activity of the motor units. If the predicted
and spontaneous commands are the same, no learning is required. If
they differ, then the difference between the spontaneous and
predicted motor commands can be used as an error signal to adjust
the weights (Fig. 3a). For instance, one could use a learning rule
known as the delta rule22, which takes the form wij = ai (a j* - a j),
where wij is the change in the weight between the presynaptic
sensory unit i and postsynaptic motor unit j, is a learning rate, a i is
the activity of the presynaptic unit, aj * is the spontaneous
postsynaptic motor activity, and a j is the predicted postsynaptic
motor activity.
This strategy works well if the sensorimotor transformation is linear—
if it can be implemented in a two-layer network (Box 1)—such as
learning to make an eye movement to a visual target. Indeed, the
retinal location of a target and the saccade vector required to acquire
that target are identical. The transformation from a sensory map (for
example, V1) to a motor map (superior colliculus) is therefore an
identity mapping, which is a linear transformation21.
Unfortunately, however, most sensorimotor transformations are
nonlinear, and the networks that compute them require at least one
intermediate layer. We showed above that a good choice for the
intermediate representation is to use basis functions. This turns out
to be a good choice for learning as well. Indeed, with basis functions,
we can decompose learning into two independent stages: first,
learning the basis functions and, second, learning the transformation
from basis functions to motor commands (Fig. 3b).
The basis functions can be learned via a purely unsupervised learning
rule. In other words, they can be learned without regard to the motor
commands being computed—before the baby even starts to move his
arm. Indeed, because by definition any basis function set can be used
to construct any motor command (Eq. 1), the choice of a basis set is
independent of the motor commands to be learned. The choice is
constrained instead by general considerations about the
computational properties of the basis functions, such as their
robustness to noise or their efficiency during learning3, as well as
considerations about biological plausibility. Gaussian and sigmoid
functions are often a good choice in both respects. It is also crucial
that the basis functions tile the entire range of input values
encountered, that is, they must form a map in the input space
considered, like the one shown in Fig. 2c. This can be done by using
variations of the Hebb and delta learning rules with an additional
term to enforce global competition, to ensure that each neuron learns
distinct basis functions23-25.
The second problem—learning the transformation from basis
functions to motor commands—is easy because motor commands are
linear combinations of basis functions (Fig. 2c). We only need to learn
the linear weights in Eq. 1, which can be done as outlined above with
a simple delta rule7, 8 (Fig. 3b). However, even a three-layer network
trained with a combination of delta rules and unsupervised learning
does not always suffice. If a nonlinear sensorimotor transformation is
a one-to-many mapping, then a network trained with delta rules will
converge on the average of all possible solutions, which is not
necessarily itself a solution. Consider a planar two-joint arm whose
segments are of equal length (Fig. 4a). There are exactly two
configurations that will reach a target lying less than two segment
lengths from the origin. A network trained using the delta rule would
converge on the average of these two configurations, which
unfortunately would overshoot the target. A similar problem would
arise in the primate arm.
Jordan and Rumelhart26 proposed a solution to this problem. They
use two networks: one to compute the required movement, and one
to map that movement into a prediction of the resulting visual
location of the hand (Fig. 4b). In engineering terms, these two
networks constitute an inverse and a forward model, respectively.
The forward model can be thought of as an internal model of the
arm; it attempts to predict the perceptual consequence of one's
action (see Wolpert and Ghahramani, this issue, for experimental
evidence supporting forward models in the brain). One role of the
forward model is to ensure that the system does not average across
commands leading to the same sensory outcome26.
Each model requires three or more layers, as both the forward and
inverse models are typically nonlinear. In principle, one can use basis
function networks and the learning strategy described above. Jordan
and Rumelhart instead used the backpropagation algorithm27, which
allows efficient propagation of the error signal through the network
layers but is not believed to be biologically plausible.
Thanks to computational modeling, we now have a better idea of how
Piaget's proposal for motor learning could be implemented in
biologically plausible neural networks using simple learning rules. The
basis function hypothesis, in particular, is well suited to the learning
of complex nonlinear sensorimotor transformations. Recent work on
the parietal cortex has suggested how these models can be tested28,
29.
Short-term memory and updating motor plans
As shown above, representations using basis functions are useful for
computing and learning sensorimotor transformations because they
provide a basis set. It turns out that these representations are also
efficient for short-term or working memory, the ability to remember a
location for a few hundred milliseconds to a few seconds. Short-term
memory is closely related to the problem of updating motor plans.
Consider a retinotopic map, that is, a population of neurons with
Gaussian tuning curves for the eye-centered location of an object. In
such a map, the appearance of an object triggers a 'hill' of activity
centered at the location of the stimulus (Fig. 5a). This hill can serve
as a memory of the object if it can be maintained over time after the
object disappears30. Much data supports the notion that this is indeed
how short-term memory is manifested in the cortex30-34. (Note that
the retinotopic map need not be arranged topographically on the
cortical surface. For example, area LIP seems to contain such a map,
inasmuch as the response fields of the neurons completely tile visual
sensory space, even though neurons coding adjacent locations do not
necessarily lie next to one another.)
To maintain a hill of activity in a network of units with Gaussian
tuning curves, one needs to add lateral connections with specific
weights. Because Gaussian tuning curves are basis function, it is
particularly easy to find the appropriate values analytically or to learn
them with biologically plausible learning procedures such as the delta
rule35. A typical solution is to use weights that generate local
excitation and long-range inhibition. Such connections seem to exist
in the cortex and are extensively used in computational modeling of
cortical circuits (for example, ref. 36). The resulting networks are
simply basis function maps to which lateral connections have been
added. Computation of motor commands and maintenance of spatial
short-term memory can therefore be done by the same basis function
neurons. This could explain why many neurons showing gain
modulation in the parietal cortex also show memory activity (L.H.S.,
unpublished data). The addition of lateral connections does not alter
in any way the computational properties of the basis function map;
indeed, lateral connections can contribute to computation of the basis
functions themselves37 or serve as optimal nonlinear noise filters38.
One requirement for a memory system in retinotopic coordinates is
that activity be updated to take into account each eye or head
movement. Consider a memory saccade task in which the target
appears 15° to the right of the fixation point and is then
extinguished. If the eyes are then moved 10° to the right before
acquiring the target, the internal representation of the target location
must be updated because the target itself is no longer visible. In this
particular case, the invisible target would now lie 5° to the right of
the fixation point. More generally, if R is the object's retinal location
and the eyes move by E, the new retinal location is R - E.
Neurophysiological evidence indicates that such updating does occur
in the brain, not only for eye-centered memory maps in response to
eye movements32, 39, 40 but also for head-centered maps in response
to head movements41.
This updating mechanism is closely related to the problem of
updating motor plans. Indeed, the eye movement required to foveate
an object is simply equal to the retinal location of the object, from
which it follows that remembering a retinal location is mathematically
equivalent to remembering an eye movement. Therefore, the
updating mechanism we have just considered can also be interpreted
as updating a motor plan for an eye movement. Interestingly,
updated memory activity is often observed in neurons that also show
presaccadic activity40, 42.
If hills of activity are used for short-term memory, updating in this
example would require moving the hill from the position
corresponding to 15° right to the position corresponding to 5° right
(Fig. 5b). As long as an eye-velocity or eye-displacement signal is
available to the network, this updating mechanism is very easy to
implement with the networks above. For instance, the lateral
connections can be modified such that the network moves the hills
with a velocity proportional to the negative of the eye velocity35.
Other ideas have been explored as well, but available experimental
data do not constrain which of these variations are used in the
brain43-45. In particular, different schemes may be used in areas with
and without topographic maps.
Discussion
Our understanding of the neural basis of sensorimotor
transformations has made outstanding progress over the last 20
years, in part because of new experimental data but also thanks to
the development of the theoretical ideas we have reviewed in this
paper. This is a prime example in which theoretical and experimental
approaches have been successfully integrated in neuroscience.
Perhaps the most remarkable observation that has come out of this
research is that seemingly distinct problems, such as computing,
learning and remembering sensorimotor transformations can be
handled by the same neural architecture. As we have seen, networks
of units with bell-shaped or sigmoidal tuning curves for sensory and
posture signals are perfectly suited to all three tasks. The key
property is that units with bell-shaped tuning curves provide basis
functions, which, when combined linearly, make it easy to compute
and learn nonlinear mappings. The simple addition of appropriate
lateral connections then adds information storage (memory) plus the
ability to update stored information after each change in posture.
Clearly, basis function networks are only the beginning of the story.
Unresolved issues abound, starting with the problem that basis
function representations are subject to combinatorial explosion, that
is, the number of neurons required increases exponentially with the
number of signals being integrated. Hence a basis function map using
10 neurons per signal and integrating 12 signals would require 1012
neurons, more than total number of neurons available in the cortex.
One solution is to use multiple modules of basis functions. For
instance, one could use two maps, connected in a hierarchical
fashion, in which the first map integrates 6 signals and the second
map the remaining 6 signals, for a total of 2 106, which is close to, if
not less than, the total number of neurons available in a single
cortical area.
It is clear that the brain indeed uses multiple cortical modules for
sensorimotor transformations, and it will be interesting to identify the
computational principles underlying this modular architecture46, 47.
Another problem will be to understand how these circuits handle
neuronal noise. Neurons are known to be noisy. This implies that we
must not only worry about how they compute but also how they do
so efficiently or reliably in the presence of noise38. These questions
promise to provide interesting issues for computational neuroscience
to address for years to come.
Received 30 May 2000; Accepted 4 October 2000.
REFERENCES
1. Tresch, M., Saltiel, P. & Bizzi, E. The construction of movement
by the spinal cord. Nat. Neurosci. 2, 162–167 (1999). MEDLINE
2. Todorov, E. Direct cortical control of muscle activation in
voluntary arm movements: a model. Nat. Neurosci. 3, 391–
398 (2000). MEDLINE
3. Poggio, T. A theory of how the brain might work. Cold Spring
Harbor Symp. Quant. Biol. 55, 899–910 (1990). MEDLINE
4. Pouget, A. & Sejnowski, T. A neural model of the cortical
representation of egocentric distance. Cereb. Cortex 4, 314–
329 (1994). MEDLINE
5. Pouget, A. & Sejnowski, T. Spatial transformations in the
parietal cortex using basis functions. J. Cogn. Neurosci. 9, 222–
237 (1997).
6. Groh, J. & Sparks, D. Two models for transforming auditory
signals from head-centered to eye-centered coordinates. Biol.
Cybern. 67, 291–302 ( 1992). MEDLINE
7. Burnod, Y. et al. Visuomotor transformations underlying arm
movements toward visual targets: a neural network model of
cerebral cortical operations. J. Neurosci. 12, 1435–1453
(1992). MEDLINE
8. Salinas, E. & Abbot, L. Transfer of coded information from
sensory to motor networks. J. Neurosci. 15, 6461–6474
(1995). MEDLINE
9. Zipser, D. & Andersen, R. A back-propagation programmed
network that stimulates reponse properties of a subset of
posterior parietal neurons . Nature 331, 679–684 (1988).
MEDLINE
10.
Andersen, R., Essick, G. & Siegel, R. Encoding of spatial
location by posterior parietal neurons . Science 230, 456–458
(1985). MEDLINE
11.
Trotter, Y., Celebrini, S., Stricanne, B., Thorpe, S. &
Imbert, M. Neural processing of stereopsis as a function of
viewing distance in primate visual area V1. J. Neurophysiol. 76,
2872–2885 ( 1997).
12.
Trotter, Y. & Celebrini, S. Gaze direction controls response
gain in primary visual-cortex neurons. Nature 398, 239–242
(1999). MEDLINE
13.
Galletti, C. & Battaglini, P. Gaze-dependent visual neurons
in area {V3a} of monkey prestriate cortex. J. Neurosci. 9,
1112–1125 (1989). MEDLINE
14.
Bremmer, F., Ilg, U., Thiele, A., Distler, C. & Hoffman, K.
Eye position effects in monkey cortex. I: Visual and pursuitrelated activity in extrastriate areas MT and MST. J.
Neurophysiol. 77, 944–961 ( 1997). MEDLINE
15.
Cumming, B. & Parker, A. Binocular neurons in V1 of
awake monkeys are selective for absolute, not relative,
disparity. J. Neurosci. 19, 5602–5618 (1999). MEDLINE
16.
Boussaoud, D., Barth, T. & Wise, S. Effects of gaze on
apparent visual responses of frontal cortex neurons. Exp. Brain
Res. 93, 423–434 (1993). MEDLINE
17.
Squatrito, S. & Maioli, M. Gaze field properties of eye
position neurones in areas MST and 7a of macaque monkey.
Vis. Neurosci. 13, 385–398 ( 1996). MEDLINE
18.
Vallar, G. Spatial hemineglect in humans. Trends Cogn.
Sci. 2, 87–97 (1998).
19.
Pouget, A., Deneve, S. & Sejnowski, T. Frames of
reference in hemineglect: a computational approach. Prog.
Brain Res. 121, 81– 97 (1999). MEDLINE
20.
Piaget, J. The Origins of Intelligence in Children (The
Norton Library, New York, 1952).
21.
Kuperstein, M. Neural model of adaptative hand-eye
coordination for single postures. Science 239, 1308–1311 (
1988). MEDLINE
22.
Widrow, B. & Hoff, M. E. in Conference proceedings of
WESCON , 96–104 (1960).
23.
Moody, J. & Darken, C. Fast learning in networks of
locally-tuned processing units. Neural Comput. 1, 281– 294
(1989).
24.
Hinton, G. & Brown, A. in Neural Information Processing
Systems vol. 12, 122–128 (MIT Press, Cambridge
Massachusetts, 2000).
25.
Olshausen, B. A. & Field, D. J. Sparse coding with an
overcomplete basis set: a strategy employed by V1? Vision Res.
37, 3311–3325 ( 1997). MEDLINE
26.
Jordan, M. & Rumelhart, D. Forward models: supervised
learning with a distal teacher. Cognit. Sci. 16, 307–354 (1990).
27.
Rumelhart, D., Hinton, G. & Williams, R. in Parallel
Distributed Processing (eds. Rumelhart, D., McClelland, J. &
Group, P. R.) 318–362 (MIT Press, Cambridge, Massachusetts,
1986).
28.
Desmurget, M. et al. Role of the posterior parietal cortex
in updating reaching movements to a visual target. Nat.
Neurosci. 2, 563–567 (1999). MEDLINE
29.
Wolpert, D. M., Goodbody, S. J. & Husain, M. Maintaining
internal representations: the role of the human superior parietal
lobe. Nat. Neurosci. 1, 529–533 (1998). MEDLINE
30.
Amit, D. The hebbian paradigm reintegrated — local
reverberations as internal representations. Behav. Brain Sci.
18, 617 –626 (1995).
31.
Fuster, J. Memory in the Cerebral Cortex: An Empirical
Approach to Neural Networks in the Human and Nonhuman
Primate (MIT Press, Cambridge, Massachusetts, 1995).
32.
Goldberg, M. & Bruce, C. Primate frontal eye fields. III.
Maintenance of a spatially accurate saccade signal. J.
Neurophysiol. 64, 489–508 (1990). MEDLINE
33.
Gnadt, J. & Mays, L. Neurons in monkey parietal area LIP
are tuned for eye-movement parameters in three-dimensional
space. J. Neurophysiol. 73, 280–297 (1995). MEDLINE
34.
Funahashi, S., Bruce, C. & Goldman-Rakic, P. Dorsolateral
prefrontal lesions and oculomotor delayed response
performance: evidence for mnemonic "scotomas". J. Neurosci.
13, 1479–1497 (1993). MEDLINE
35.
Zhang, K. Representation of spatial orientation by the
intrinsic dynamics of the head-direction cell ensemble: a
theory. J. Neurosci. 16, 2112–2126 (1996). MEDLINE
36.
Somers, D. C., Nelson, S. B. & Sur, M. An emergent
model of orientation selectivity in cat visual cortical simple cells.
J. Neurosci. 15, 5448–5465 (1995). MEDLINE
37.
Salinas, E. & Abbott, L. F. A model of multiplicative neural
responses in parietal cortex. Proc. Natl. Acad. Sci. USA 93,
11956–11961 (1996). MEDLINE
38.
Deneve, S., Latham, P. & Pouget, A. Reading population
codes: A neural implementation of ideal observers. Nat.
Neurosci. 2, 740– 745 (1999). MEDLINE
39.
Walker, M., Fitzgibbon, E. & Goldberg, M. Neurons in the
monkey superior colliculus predict the visual result of
impending saccadic eye movements. J. Neurophysiol. 73,
1988–2003 ( 1995). MEDLINE
40.
Mazzoni, P., Bracewell, R., Barash, S. & Andersen, R.
Motor intention activity in the macaque's lateral intraparietal
area. I. Dissociation of motor plan from sensory memory. J.
Neurophysiol. 76, 1439–1456 (1996). MEDLINE
41.
Graziano, M., Hu, X. & Gross, C. Coding the locations of
objects in the dark. Science 277, 239–241 (1997). MEDLINE
42.
Duhamel, J. R., Colby, C. L. & Goldberg, M. E. The
updating of the representation of visual space in parietal cortex
by intended eye movements. Science 255, 90–92 (1992).
MEDLINE
43.
Droulez, J. & Berthoz, A. A neural model of sensoritopic
maps with predictive short-term memory properties. Proc. Natl.
Acad. Sci. USA 88, 9653–9657 ( 1991). MEDLINE
44.
Dominey, P. & Arbib, M. A cortico-subcortical model for
the generation of spatially accurate sequential saccades. Cereb.
Cortex 2, 153–175 ( 1992). MEDLINE
45.
Seung, H. How the brain keeps the eyes still. Proc. Natl.
Acad. Sci. USA 93, 13339–13344 ( 1996). MEDLINE
46.
Snyder, L., Batista, A. & Andersen, R. Coding of intention
in the posterior parietal cortex . Nature 386, 167–170 (1997).
MEDLINE
47.
Snyder, L., Grieve, K., Brotchie, P. & Andersen, R.
Separate body- and world-referenced representations of visual
space in parietal cortex. Nature 394, 887–891 ( 1998).
MEDLINE
Figure 1: Linear and nonlinear functions.
(a) The linear function z = 2x + 3y. In general, a function is
linear if it can be written as a weighted sum of its input
variables plus a constant. All other functions are nonlinear.
Linear functions form lines (one variable), planes (two
variables, as shown here) or hyperplanes (more than two
variables). (b) Linear functions can be implemented in twolayer networks. The network shown corresponds to the
linear function in (a). (c) The nonlinear function z = exp(-(
x2 + y2 - 100) 2/1000). Nonlinear functions are not planar
and can form surfaces that can be arbitrarily complex, such
as a circular ridge. (d) A neural network implementation of
the nonlinear function in (c) using Gaussian basis functions
in the intermediate representation (Box 1). The basis
function units are organized so as to form a map in the x- y
plane. Right, two representative response functions of these
basis function units. The activity of the output unit is
obtained by taking a linear sum of the basis function units.
In this example, the weights of the blue units onto the
output unit are set to one, whereas all the other units have a
weight of zero. As a result, the output unit mathematically
sums a set of Gaussian functions arranged along a circle in
the x-y plane. This leads to the response function (right),
which is similar to the circular ridge in (c).
Figure 2: Basis function units.
(a) The response function of a basis function unit computing
a product of a Gaussian function of retinal location (eyecentered position) multiplied by a sigmoidal function of eye
position. (b) A mapping of the retinotopic receptive field
derived from a unit with the properties in (a) for three
different eye positions. Bold lines in (a) correspond to the
three curves shown here. The receptive field always peaks at
the same retinal location, but the gain (or amplitude) of the
response varies with eye position. Gain modulations similar
to this are found in many cortical areas, from V1 to the
premotor cortex. (c) A neural network model for nonlinear
sensorimotor transformations using basis functions. The
input layers encode the retinal location of an object and the
current eye position, whereas the output layer encodes the
change in joint angles of the arm. Other inputs signals are
needed to compute the change in joint angles of the arm,
such as head position, starting hand location and so on, but
for simplicity we show only the visual and eye position
inputs. The tuning curve of the eye-centered and eyeposition units are assumed to follow Gaussian and sigmoid
functions, respectively. This nonlinear sensorimotor
transformation requires an intermediate layer. In the case
illustrated here, the intermediate layer uses basis function
units. Each unit computes the product of the activity of one
input unit from the eye-centered map and one input unit
from the eye position map. This leads to the response of the
basis function units (a, b). The transformation from the
basis function to the output units involves a simple linear
transformation, namely, a weighted sum of the activity of
the basis functions units. This is the main advantage of this
approach: once the basis functions are computed, nonlinear
transformations become linear.
Figure 3: Learning sensorimotor transformations in
neural networks.
(a) Learning motor commands with spontaneous
movements. The motor layer generates a spontaneous
pattern of activity. This activity is fed into the arm, resulting
in a new arm position. The position of the arm is observed in
the input sensory layer. The sensory activity is passed
through the weights to compute a predicted motor
command. The weights can then be adjusted according to an
error signal obtained by computing the difference between
the spontaneous and predicted motor command. By
repeating this procedure many times, one can learn the
appropriate sensorimotor mapping. Note that the same
motor units are involved in all steps: generating and
remembering the spontaneous motor command, computing
the predicted motor command and computing the error
signal. In most models, the neuronal mechanisms underlying
these various steps are not specified; the motor units are
simply assumed to be able to perform the required
computations. (b) When the transformation is nonlinear, as
is the case for arm movements, an intermediate layer of
units is required. If basis functions are used in the
intermediate layer, learning can be done in two stages. The
weights to the basis function can be learned with an
unsupervised, or self-organizing rule because the basis
functions only depend on the input, not on the motor
command computed in the output layer. The weights to the
output layer are equivalent to the ones in the network in (a)
and can be learned using the same error signal.
Figure 4: One-to-many sensorimotor transformations.
(a) To reach for the location indicated by the star, a twojoint arm working in a two-dimensional plane can adopt
exactly two configurations (top). The transformation in this
case is from one sensory input into one of two motor
commands. In the case illustrated here, the two arm
configurations are mirror images of each other, that is, the
joint angles have the same amplitude but reversed signs. As
a result, the average of angle 1 and 2 over the two
configurations is zero. The average configuration (bottom)
would therefore overshoot the target (red arrow). Because
averaging is precisely what the network in Fig. 3b does
during training, this transformation cannot be learned
properly with this kind of networks. (b) One solution is to
use two networks, one for the sensorimotor transformation
(the inverse model) and one for predicting the sensory
consequences of the motor command (the forward model).
Each of these networks typically involves three or more
layers such as the one shown in Fig. 3b. Learning in this
system is achieved by learning the forward model (for
example, through the generation of spontaneous motor
commands) and then using the forward model to train the
inverse model.
Figure 5: Short-term memory networks for
sensorimotor transformations.
(a) A hill of activity (top) in a network organized in a twodimensional retinotopic map (bottom). If the hill is a stable
state for the network (that is, if it maintains itself over
time), then the position of the peak of the hill can be used
as a memory for retinal location. In the case illustrated here,
the network is storing the value (15°, 0°), as indicated by
the arrows. (b) To remain spatially accurate, a memory of
the retinal location of an object must be updated after each
eye movement. This updating can take place in such
memory networks by displacing the hill by an amount equal
to minus the eye displacement (- E). The response to a
+10° horizontal eye movement is shown.